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A274471 Numbers missing from A134419 despite satisfying the necessary congruence conditions (see comments). 4
564, 842, 1284, 2306, 2308, 2402, 2459, 3602, 3650, 3803, 6242, 6338, 6779, 7044, 7058, 7319, 7643, 8088, 8354, 8363, 8402, 8543, 8628, 9122, 9168, 9412, 10607, 10826, 10852, 11257, 11378, 11447, 12203, 12436, 12458, 12722, 12984, 13682, 14162, 14388, 14424, 14639 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A134419 consists of those n where x^2 - n*y^2 = n(n-1)(n+1)/3 has integer solutions for x and y. There are easily verified necessary congruence conditions for that to occur:

  (defining x||y to mean x|y and x and y/x are coprime)

  if 3^e||n with e>0, then e is odd and (n/3^e)=2 (mod 3);

  if p^e||n with p=5 or 7 (mod 12), then e is even;

  if 3^e||(n+1) with e>0, then e is odd;

  if p^e||(n+1) with p=3 (mod 4) and p>3, then e is even.

However, these conditions are not sufficient. This sequence consists of the numbers n satisfying the congruence conditions but for which the Pellian equation has no integer solutions.

If n = k^2*m where m is squarefree, then a necessary (but not sufficient) condition for n to occur in this sequence is that the narrow class group of quadratic forms of discriminant 4*m has more than one class per genus, or equivalently that the narrow class group is not an elementary 2-group.

LINKS

Christopher E. Thompson, Table of n, a(n) for n = 1..799 [values up to 250000]

CROSSREFS

Cf. A134419, A274469, A274470.

Sequence in context: A237029 A183732 A014361 * A252293 A252286 A252285

Adjacent sequences:  A274468 A274469 A274470 * A274472 A274473 A274474

KEYWORD

nonn

AUTHOR

Christopher E. Thompson, Jun 24 2016

STATUS

approved

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Last modified January 20 03:25 EST 2022. Contains 350467 sequences. (Running on oeis4.)